international journal of pure and applied mathematics · many applications to geometry (as for...

International Journal of Pure and Applied Mathematics————————————————————————–Volume 48 No. 4 2008, 451-472

ON DIFFERENTIABILITY AND INTEGRABILITY

OF HYPERCOMPLEX OCTONIONIC FUNCTIONS

C.A. Pendeza1, M.F. Borges2 §, J.M. Machado3

1,2,3Department of ComputingUNESP - Sao Paulo State University

S.J. Rio Preto Campus, 15054-000 - Sao Jose do Rio Preto, BRAZIL1e-mail: [email protected]: [email protected]: [email protected]

Abstract: In this paper we revisit some results obtained recently by two ofus (Borges and Machado [13], [14]) for quaternion functions, in order to obtainCauchy-Riemann like relations, “hipercomplex derivatives” and “hypercomplexintegrals” for a class of functions over octonion numbers.

AMS Subject Classification: 30G99, 30E99Key Words: hipercomplex functions, hipercomplex derivatives, hypercom-plex integrals

1. Introduction

Hypercomplex numbers like quaternions and octonions have been used in physicssince Hamilton’s and Graves’ earlier works [10]. Examples of quaternion ap-plications may be found in Lorentz transformations in special relativity (DeLeo [15] and Rastall [5]), in Dirac and Klein-Gordon equations (Calixto [2]), inmany applications to geometry (as for instance in R. Graves [8] and Hankins[11]), as well as in Maxwell’s equations (Colomb [4]). Other hypercomplex likethe octonions may also give a sound support not only for physical theories, aswell as set up analogies to the theory of analytic functions (Gursey [9]), to theconstruction of vertex operator algebras [6], and of symmetric spaces (Besse

Received: March 12, 2008 c© 2008, Academic Publications Ltd.

§Correspondence author

452 C.A. Pendeza, M.F. Borges, J.M. Machado

[1]), and in many other approaches and applications to physics (Catto [3]). Inanalysing some of the recent results on differentiability of quaternionionic func-tions (Borges and Machado [13], [14]), it is shown how Cauchy-Riemann likerelations associated with these differentiability criteria may define conformalmappings acting on superfaces of the Euclidian four dimensional space. In thispaper we consider these results for quaternionic formulation, and generalizethem for a class of hypercomplex octonionic functions.

2. Brief Comments on The Octonions Algebra

The octonions are a somewhat nonassociativite extension of the quaternions.They form the 8-dimensional normed division algebra on R.

The octonionic algebra, also called octaves denoted for O, is an alternativedivision algebra.

The octonions set O is denoted by

O = {a, b, c, d, e, f, g, h} ∈ R,

where

(a, b, c, d, e, f, g, h) = (a′, b′, c′, d′, e′, f ′, g′, h′) ⇐⇒ a = a′, b = b′, c = c′, d = d′,

e = e′, f = f ′, g = g′, h = h′.

The octonions do not form a ring due the non-commutativity of the multi-plication. Also do not form a group due a nonassociativide of multiplication.They form a Moufang loop, a loop with identity element (N. Jacobson, [12]).

Let us consider an octonionic number given by

o = a + bi + cj + dk + el + f li + glj + hlk .

The octonion unities 1, i, j, k, l, li, lj, lk form an orthonormal base of the8-dimensional algebra. The multiplications of the base elements are obtainedthrough of the multiplication board (Table 1).

3. On the Differentiability of Hypercomplex Octonionic Functions

The first attempt in order to reach a generalization from the 2-dimensionalcomplex to a 4-dimensional hypercomplex theory was realized by R. Fueter [7]in the decade of 1930, through the concept of left and right hypercomplex reg-ularity. He showed for first time an analogue of Cauchy-Riemann relations for

ON DIFFERENTIABILITY AND INTEGRABILITY... 453

∗ 1 i j k l li lj lk

1 1 i j k l li lj lk

i i −1 k −j −li l −lk lj

j j −k −1 i −lj lk l −li

k k j −i −1 −lk −lj li l

l l li lj lk −1 −i −j −k

li li −l −lk lj i −1 −k j

lj lj lk −l −li j k −1 −i

lk lk −lj li −l k −j i −1

(1)

Table 1: The octonions multiplication table

quaternions. We extend such results for octonions, following a path establishedin Borges and Machado [13], [14].

By definition, let E8 ⊂ O be an 8-dimensional Euclidian space and a hyper-complex octonionic number o = u1+u2i+u3j+u4k+u5l+u6li+u7lj+u8lk ∈ R8.

Thus, a hypercomplex octonionic function f : E8 → O is a map that makes acorrespondence between each o ∈ E8 to an octonionic function w = f(o). Thefunction f constructed as

f : E8 → O

(u1, u2, u3, u4, u5, u6, u7, u8) 7→ f(u1, u2, u3, u4, u5, u6, u7, u8),

where,

f(u1, u2, u3, u4, u5, u6, u7, u8) = f1(u1, u2, u3, u4, u5, u6, u7, u8)

+ f2(u1, u2, u3, u4, u5, u6, u7, u8)i

+ f3(u1, u2, u3, u4, u5, u6, u7, u8)j

+ f4(u1, u2, u3, u4, u5, u6, u7, u8)k

+ f5(u1, u2, u3, u4, u5, u6, u7, u8)l

+ f6(u1, u2, u3, u4, u5, u6, u7, u8)li

+ f7(u1, u2, u3, u4, u5, u6, u7, u8)lj

+ f8(u1, u2, u3, u4, u5, u6, u7, u8)lk (2)

and i, j, k, l, li, lj and lk obey the corresponding multiplication Table 1:fl : R8 → R are real-valued coordinate functions. The hypercomplex variable,denoted as o, is given by

o = u1 + u2i + u3j + u4k + u5l + u6li + u7lj + u8lk, (3)


where ui ∈ R. By taking the multiplication rules (Table 1) for octonions’quasegroup, they allow a formal definition of

∫

fdz as

∫ b

a

fdz =

∫ b

a

(f1 + f2i + f3j + f4k + f5l + f6li + f7lj + f8lk)

(du1 + du2i + du3j + du4k + du5l + du6li + du7lj + du8lk)

=

∫

b

a

(f1du1 − f2du2 − f3du3 − f4du4 − f5du5 − f6du6 − f7du7 − f8du8)

+

∫

b

a

(f2du1 + f1du2 − f4du3 + f3du4 + f6du5 − f5du6 + f8du7 − f7du8)i

+

∫ b

a

(f3du1 + f4du2 + f1du3 − f2du4 + f7du5 − f8du6 − f5du7 + f6du8)j

+

∫

b

a

(f4du1 − f3du2 + f2du3 + f1du4 + f8du5 + f7du6 − f6du7 − f5du8)k

+

∫

b

a

(f5du1 − f6du2 − f7du3 − f8du4 + f1du5 + f2du6 + f3du7 + f4du8)l

+

∫ b

a

(f6du1 + f5du2 + f8du3 − f7du4 − f2du5 + f1du6 + f4du7 − f3du8)li

+

∫ b

a

(f7du1 − f8du2 + f5du3 + f6du4 − f3du5 − f4du6 + f1du7 + f2du8)lj

+

∫

b

a

(f8du1 + f7du2 − f6du3 + f5du4 − f4du5 + f3du6 − f2du7 + f1du8)lk. (4)

Let us suppose first that the coordinate functions fl : R8 → R satisfy the re-quirements for the existence of all integrals in (3), and given a path from a pointa = (a1, a2, a3, a4, a5, a6, a7, a8) to a point b = (b1, b2, b3, b4, b5, b6, b7, b8) in a

simply connected domain of the eight-dimensional space, the integral∫ b

af(z)dz

is independent from the integration path under the conditions of the followingtheorem.

Theorem 1. For any pair of points a and b and any path joining them

in a simply connected subdomain of the eight-dimensional space, the integral∫ b

afdz is independent from the given path if and only if there is a functions

F = F1+F2i+F3j+F4k+F5l+F6li+F7lj+F8lk such that∫ b

afdz = F (b)−F (a),

and satisfying the following relations:

∂F1

∂u1=

∂F2

∂u2=

∂F3

∂u3=

∂F4

∂u4=

∂F5

∂u5=

∂F6

∂u6=

∂F7

∂u7=

∂F8

∂u8,


∂F2

∂u1= −

∂F1

∂u2=

∂F4

∂u3= −

∂F3

∂u4= −

∂F6

∂u5=

∂F5

∂u6= −

∂F8

∂u7=

∂F7

∂u8,

∂F3

∂u1= −

∂F4

∂u2= −

∂F1

∂u3=

∂F2

∂u4= −

∂F7

∂u5=

∂F8

∂u6=

∂F5

∂u7= −

∂F6

∂u8,

∂F4

∂u1=

∂F3

∂u2= −

∂F2

∂u3= −

∂F1

∂u4= −

∂F8

∂u5= −

∂F7

∂u6=

∂F6

∂u7=

∂F5

∂u8,

∂F5

∂u1=

∂F6

∂u2=

∂F7

∂u3=

∂F8

∂u4= −

∂F1

∂u5= −

∂F2

∂u6= −

∂F3

∂u7= −

∂F4

∂u8,

∂F6

∂u1= −

∂F5

∂u2= −

∂F8

∂u3=

∂F7

∂u4=

∂F2

∂u5= −

∂F1

∂u6= −

∂F4

∂u7=

∂F3

∂u8,

∂F7

∂u1=

∂F8

∂u2= −

∂F5

∂u3= −

∂F6

∂u4=

∂F3

∂u5=

∂F4

∂u6= −

∂F1

∂u7= −

∂F2

∂u8,

∂F8

∂u1= −

∂F7

∂u2=

∂F6

∂u3= −

∂F5

∂u4=

∂F4

∂u5= −

∂F3

∂u6=

∂F2

∂u7= −

∂F1

∂u8. (5)

Proof. According to (4), the path integral is written as

=

∫

b

a

(f1du1 − f2du2 − f3du3 − f4du4 − f5du5 − f6du6 − f7du7 − f8du8)

+

∫ b

a

(f2du1 + f1du2 − f4du3 + f3du4 + f6du5 − f5du6 + f8du7 − f7du8)i

+

∫ b

a

(f3du1 + f4du2 + f1du3 − f2du4 + f7du5 − f8du6 − f5du7 + f6du8)j

+

∫

b

a

(f4du1 − f3du2 + f2du3 + f1du4 + f8du5 + f7du6 − f6du7 − f5du8)k

+

∫ b

a

(f5du1 − f6du2 − f7du3 − f8du4 + f1du5 + f2du6 + f3du7 + f4du8)l

+

∫ b

a

(f6du1 + f5du2 + f8du3 − f7du4 − f2du5 + f1du6 + f4du7 − f3du8)li

+

∫

b

a

(f7du1 − f8du2 + f5du3 + f6du4 − f3du5 − f4du6 + f1du7 + f2du8)lj

+

∫ b

a

(f8du1 + f7du2 − f6du3 + f5du4 − f4du5 + f3du6 − f2du7 + f1du8)lk,

and is independent from the path, since

∫ b

a

fdz =

∫ b

a

dF =

∫ b

a

d(F1 + F2i + F3j + F4k + F5l + F6li + F7lj + F8lk)


= F (b) − F (a), (6)

for a certain function F . Supposing that such a function F there exists, thetotal differentials of its coordinate functions are given by

dF1 =∂F1

∂u1

du1+∂F1

∂u2

du2+∂F1

∂u3

du3+∂F1

∂u4

du4+∂F1

∂u5

du5+∂F1

∂u6

du6+∂F1

∂u7

du7+∂F1

∂u8

du8

= f1du1 − f2du2 − f3du3 − f4du4 − f5du5 − f6du6 − f7du7 − f8du8, (7)

dF2 =∂F2

∂u1

du1+∂F2

∂u2

du2+∂F2

∂u3

du3+∂F2

∂u4

du4+∂F2

∂u5

du5+∂F2

∂u6

du6+∂F2

∂u7

du7+∂F2

∂u8

du8

= f2du1 + f1du2 − f4du3 + f3du4 + f6du5 − f5du6 + f8du7 − f7du8, (8)

dF3 =∂F3

∂u1

du1+∂F3

∂u2

du2+∂F3

∂u3

du3+∂F3

∂u4

du4+∂F3

∂u5

du5+∂F3

∂u6

du6+∂F3

∂u7

du7+∂F3

∂u8

du8

= f3du1 + f4du2 + f1du3 − f2du4 + f7du5 − f8du6 − f5du7 + f6du8, (9)

dF4 =∂F4

∂u1

du1+∂F4

∂u2

du2+∂F4

∂u3

du3+∂F4

∂u4

du4+∂F4

∂u5

du5+∂F4

∂u6

du6+∂F4

∂u7

du7+∂F4

∂u8

du8

= f4du1 − f3du2 + f2du3 + f1du4 + f8du5 + f7du6 − f6du7 − f5du8, (10)

dF5 =∂F5

∂u1

du1+∂F5

∂u2

du2+∂F5

∂u3

du3+∂F5

∂u4

du4+∂F5

∂u5

du5+∂F5

∂u6

du6+∂F5

∂u7

du7+∂F5

∂u8

du8

= f5du1 − f6du2 − f7du3 − f8du4 + f1du5 + f2du6 + f3du7 + f4du8, (11)

dF6 =∂F6

∂u1

du1+∂F6

∂u2

du2+∂F6

∂u3

du3+∂F6

∂u4

du4+∂F6

∂u5

du5+∂F6

∂u6

du6+∂F6

∂u7

du7+∂F6

∂u8

du8

= f6du1 + f5du2 + f8du3 − f7du4 − f2du5 + f1du6 + f4du7 − f3du8, (12)

dF7 =∂F7

∂u1

du1+∂F7

∂u2

du2+∂F7

∂u3

du3+∂F7

∂u4

du4+∂F7

∂u5

du5+∂F7

∂u6

du6+∂F7

∂u7

du7+∂F7

∂u8

du8

= f7du1 − f8du2 + f5du3 + f6du4 − f3du5 − f4du6 + f1du7 + f2du8, (13)

dF8 =∂F8

∂u1

du1+∂F8

∂u2

du2+∂F8

∂u3

du3+∂F8

∂u4

du4+∂F8

∂u5

du5+∂F8

∂u6

du6+∂F8

∂u7

du7+∂F8

∂u8

du8

= f8du1 + f7du2 − f6du3 + f5du4 − f4du5 + f3du6 − f2du7 + f1du8, (14)

and therefore the relations of Theorem 1 are immediate:

f1 =∂F1

∂u1

=∂F2

∂u2

=∂F3

∂u3

=∂F4

∂u4

=∂F5

∂u5

=∂F6

∂u6

=∂F7

∂u7

=∂F8

∂u8

,


f2 =∂F2

∂u1

= −∂F1

∂u2

=∂F4

∂u3

= −∂F3

∂u4

= −∂F6

∂u5

=∂F5

∂u6

= −∂F8

∂u7

=∂F7

∂u8

,

f3 =∂F3

∂u1

= −∂F4

∂u2

= −∂F1

∂u3

=∂F2

∂u4

= −∂F7

∂u5

=∂F8

∂u6

=∂F5

∂u7

= −∂F6

∂u8

,

f4 =∂F4

∂u1

=∂F3

∂u2

= −∂F2

∂u3

= −∂F1

∂u4

= −∂F8

∂u5

= −∂F7

∂u6

=∂F6

∂u7

=∂F5

∂u8

,

f5 =∂F5

∂u1

=∂F6

∂u2

=∂F7

∂u3

=∂p8

∂u4

= −∂F1

∂u5

= −∂F2

∂u6

= −∂F3

∂u7

= −∂F4

∂u8

,

f6 =∂F6

∂u1

= −∂F5

∂u2

= −∂F8

∂u3

=∂F7

∂u4

=∂F2

∂u5

= −∂F1

∂u6

= −∂F4

∂u7

=∂F3

∂u8

,

f7 =∂F7

∂u1

=∂F8

∂u2

= −∂F5

∂u3

= −∂F6

∂u4

=∂F3

∂u5

=∂F4

∂u6

= −∂F1

∂u7

= −∂F2

∂u8

,

f8 =∂F8

∂u1

= −∂F7

∂u2

=∂F6

∂u3

= −∂F5

∂u4

=∂F4

∂u5

= −∂F3

∂u6

=∂F2

∂u7

= −∂F1

∂u8

.

Lemma 1. Given a function f(z) over the octonionic quasegroup O, withproperly differentiable coordinate functions satisfying the relations of Theorem1,

∂f1

∂u1

=∂f2

∂u2

=∂f3

∂u3

=∂f4

∂u4

=∂f5

∂u5

=∂F6

∂u6

=∂f7

∂u7

=∂f8

∂u8

,

∂f2

∂u1

= −∂f1

∂u2

=∂f4

∂u3

= −∂f3

∂u4

= −∂f6

∂u5

=∂f5

∂u6

= −∂f8

∂u7

=∂f7

∂u8

,

∂f3

∂u1

= −∂f4

∂u2

= −∂f1

∂u3

=∂f2

∂u4

= −∂f7

∂u5

=∂f8

∂u6

=∂f5

∂u7

= −∂f6

∂u8

,

∂f4

∂u1

=∂f3

∂u2

= −∂f2

∂u3

= −∂f1

∂u4

= −∂f8

∂u5

= −∂f7

∂u6

=∂f6

∂u7

=∂f5

∂u8

,

∂f5

∂u1

=∂f6

∂u2

=∂f7

∂u3

=∂f8

∂u4

= −∂f1

∂u5

= −∂f2

∂u6

= −∂f3

∂u7

= −∂f4

∂u8

,


∂f6

∂u1

= −∂f5

∂u2

= −∂f8

∂u3

=∂f7

∂u4

=∂f2

∂u5

= −∂f1

∂u6

= −∂f4

∂u7

=∂f3

∂u8

,

∂f7

∂u1

=∂f8

∂u2

= −∂f5

∂u3

= −∂f6

∂u4

=∂f3

∂u5

=∂f4

∂u6

= −∂f1

∂u7

= −∂f2

∂u8

,

∂f8

∂u1

= −∂f7

∂u2

=∂f6

∂u3

= −∂f5

∂u4

=∂f4

∂u5

= −∂f3

∂u6

=∂f2

∂u7

= −∂f1

∂u8

, (15)

and a function g(z) defined in terms of f(z) by

g(z) =1

8

[(

∂f1

∂u1+

∂f2

∂u2+

∂f3

∂u3+

∂f4

∂u4+

∂f5

∂u5+

∂f6

∂u6+

∂f7

∂u7+

∂f8

∂u8

)

+

(

∂f2

∂u1−

∂f1

∂u2−

∂f4

∂u3+

∂f3

∂u4+

∂f6

∂u5−

∂f5

∂u6+

∂f8

∂u7−

∂f7

∂u8

)

i

+

(

∂f3

∂u1+

∂f4

∂u2−

∂f1

∂u3−

∂f2

∂u4+

∂f7

∂u5−

∂f8

∂u6−

∂f5

∂u7+

∂f6

∂u8

)

j

+

(

∂f4

∂u1−

∂f3

∂u2+

∂f2

∂u3−

∂f1

∂u4+

∂f8

∂u5+

∂f7

∂u6−

∂f6

∂u7−

∂f5

∂u8

)

k

+

(

∂f5

∂u1−

∂f6

∂u2−

∂f7

∂u3−

∂f8

∂u4−

∂f1

∂u5+

∂f2

∂u6+

∂f3

∂u7+

∂f4

∂u8

)

l

+

(

∂f6

∂u1+

∂f5

∂u2+

∂f8

∂u3−

∂f7

∂u4−

∂f2

∂u5−

∂f1

∂u6+

∂f4

∂u7−

∂f3

∂u8

)

li

+

(

∂f7

∂u1−

∂f8

∂u2+

∂f5

∂u3+

∂f6

∂u4−

∂f3

∂u5−

∂f4

∂u6−

∂f1

∂u7+

∂f2

∂u8

)

lj

+

(

∂f8

∂u1+

∂f7

∂u2−

∂f6

∂u3+

∂f5

∂u4−

∂f4

∂u5+

∂f3

∂u6−

∂f2

∂u7−

∂f1

∂u8

)

lk

]

,

then∫

g(z)dz = f(z), and hence g(z) may formally be treated as the “left-

octonion” derivative of f(z) and denoted by g(z) = dfr(z)dz

.

Proof. By taking again the multiplication Table 1 for the octonions’ quase-group and writing g(z)dz, then

g(z)dz =1

8

(

∂f1

∂u1

+∂f2

∂u2

+∂f3

∂u3

+∂f4

∂u4

+∂f5

∂u5

+∂f6

∂u6

+∂f7

∂u7

+∂f8

∂u8

)

du1


−1

8

(

∂f2

∂u1

−∂f1

∂u2

+∂f4

∂u3

−∂f3

∂u4

−∂f6

∂u5

+∂f5

∂u6

−∂f8

∂u7

+∂f7

∂u8

)

du2

−1

8

(

∂f3

∂u1

−∂f4

∂u2

−∂f1

∂u3

+∂f2

∂u4

−∂f7

∂u5

+∂f8

∂u6

+∂f5

∂u7

−∂f6

∂u8

)

du3

−1

8

(

∂f4

∂u1

+∂f3

∂u2

−∂f2

∂u3

−∂f1

∂u4

−∂f8

∂u5

−∂f7

∂u6

+∂f6

∂u7

+∂f5

∂u8

)

du4

−1

8

(

∂f5

∂u1

+∂f6

∂u2

+∂f7

∂u3

+∂f8

∂u4

−∂f1

∂u5

−∂f2

∂u6

−∂f3

∂u7

−∂f4

∂u8

)

du5

−1

8

(

∂f6

∂u1

−∂f5

∂u2

−∂f8

∂u3

+∂f7

∂u4

+∂f2

∂u5

−∂f1

∂u6

−∂f4

∂u7

+∂f3

∂u8

)

du6

−1

8

(

∂f7

∂u1

+∂f8

∂u2

−∂f5

∂u3

−∂f6

∂u4

+∂f3

∂u5

+∂f4

∂u6

−∂f1

∂u7

−∂f2

∂u8

)

du7

−1

8

(

∂f8

∂u1

−∂f7

∂u2

+∂f6

∂u3

−∂f5

∂u4

+∂f4

∂u5

−∂f3

∂u6

+∂f2

∂u7

−∂f1

∂u8

)

du8

+1

8i

[(

∂f2

∂u1

−∂f1

∂u2

+∂f4

∂u3

−∂f3

∂u4

−∂f6

∂u5

+∂f5

∂u6

−∂f8

∂u7

+∂f7

∂u8

)

du1

+

(

∂f1

∂u1

+∂f2

∂u2

+∂f3

∂u3

+∂f4

∂u4

+∂f5

∂u5

+∂f6

∂u6

+∂f7

∂u7

+∂f8

∂u8

)

du2

+

(

∂f4

∂u1

+∂f3

∂u2

−∂f2

∂u3

−∂f1

∂u4

−∂f8

∂u5

−∂f7

∂u6

+∂f6

∂u7

+∂f5

∂u8

)

du3

−

(

∂f3

∂u1

−∂f4

∂u2

−∂f1

∂u3

+∂f2

∂u4

−∂f7

∂u5

+∂f8

∂u6

+∂f5

∂u7

−∂f6

∂u8

)

du4


−

(

∂f6

∂u1

−∂f5

∂u2

−∂f8

∂u3

+∂f7

∂u4

+∂f2

∂u5

−∂f1

∂u6

−∂f4

∂u7

−∂f3

∂u8

)

du5

+

(

∂f5

∂u1

+∂f6

∂u2

+∂f7

∂u3

+∂f8

∂u4

−∂f1

∂u5

−∂f2

∂u6

−∂f3

∂u7

−∂f4

∂u8

)

du6

−

(

∂f8

∂u1

−∂f7

∂u2

+∂f6

∂u3

−∂f5

∂u4

+∂f4

∂u5

−∂f3

∂u6

+∂f2

∂u7

−∂f1

∂u8

)

du7

+

(

∂f5

∂u1

+∂f6

∂u2

+∂f7

∂u3

+∂f8

∂u4

−∂f1

∂u5

−∂f2

∂u6

−∂f3

∂u7

−∂f4

∂u8

)

du8

]

+1

8j

[(

∂f3

∂u1

−∂f4

∂u2

−∂f1

∂u3

+∂f2

∂u4

−∂f7

∂u5

+∂f8

∂u6

+∂f5

∂u7

−∂f6

∂u8

)

du1

−

(

∂f4

∂u1

+∂f3

∂u2

−∂f2

∂u3

−∂f1

∂u4

−∂f8

∂u5

−∂f7

∂u6

+∂f6

∂u7

+∂f5

∂u8

)

du2

+

(

∂f1

∂u1

+∂f2

∂u2

+∂f3

∂u3

+∂f4

∂u4

+∂f5

∂u5

+∂f6

∂u6

+∂f7

∂u7

+∂f8

∂u8

)

du3

+

(

∂f2

∂u1

−∂f1

∂u2

+∂f4

∂u3

−∂f3

∂u4

−∂f6

∂u5

+∂f5

∂u6

−∂f8

∂u7

+∂f7

∂u8

)

du4

−

(

∂f7

∂u1

+∂f8

∂u2

−∂f5

∂u3

−∂f6

∂u4

+∂f3

∂u5

+∂f4

∂u6

−∂f1

∂u7

−∂f2

∂u8

)

du5

+

(

∂f8

∂u1

−∂f7

∂u2

+∂f6

∂u3

−∂f5

∂u4

+∂f4

∂u5

−∂f3

∂u6

+∂f2

∂u7

−∂f1

∂u8

)

du6

+

(

∂f5

∂u1

+∂f6

∂u2

+∂f7

∂u3

+∂f8

∂u4

−∂f1

∂u5

−∂f2

∂u6

−∂f3

∂u7

−∂f3

∂u8

)

du7


−

(

∂f6

∂u1

−∂f5

∂u2

−∂f8

∂u3

+∂f7

∂u4

+∂f2

∂u5

−∂f1

∂u6

−∂f4

∂u7

+∂f3

∂u8

)

du8

]

+1

8k

[(

∂f4

∂u1

+∂f3

∂u2

−∂f2

∂u3

−∂f1

∂u4

−∂f8

∂u5

−∂f7

∂u6

+∂f6

∂u7

+∂f5

∂u8

)

du1

+

(

∂f3

∂u1

−∂f4

∂u2

−∂f1

∂u3

+∂f2

∂u4

−∂f7

∂u5

+∂f8

∂u6

+∂f5

∂u7

−∂f6

∂u8

)

du2

−

(

∂f2

∂u1

−∂f1

∂u2

+∂f4

∂u3

−∂f3

∂u4

−∂f6

∂u5

+∂f5

∂u6

−∂f8

∂u7

+∂f7

∂u8

)

du3

+

(

∂f1

∂u1

+∂f2

∂u2

+∂f3

∂u3

+∂f4

∂u4

+∂f5

∂u5

+∂f6

∂u6

+∂f7

∂u7

+∂f8

∂u8

)

du4

−

(

∂f8

∂u1

−∂f7

∂u2

+∂f6

∂u3

−∂f5

∂u4

+∂f4

∂u5

−∂f3

∂u6

+∂f2

∂u7

−∂f1

∂u8

)

du5

−

(

∂f7

∂u1

+∂f8

∂u2

−∂f5

∂u3

−∂f6

∂u4

+∂f3

∂u5

+∂f4

∂u6

−∂f1

∂u7

−∂f2

∂u8

)

du6

+

(

∂f6

∂u1

−∂f5

∂u2

−∂f8

∂u3

+∂f7

∂u4

+∂f2

∂u5

−∂f1

∂u6

−∂f4

∂u7

+∂f3

∂u8

)

du7

−

(

∂f5

∂u1

+∂f6

∂u2

+∂f7

∂u3

+∂f8

∂u4

−∂f1

∂u5

−∂f2

∂u6

−∂f3

∂u7

−∂f4

∂u8

)

du8

]

+1

8l

[(

∂f5

∂u1

+∂f6

∂u2

+∂f7

∂u3

+∂f8

∂u4

−∂f1

∂u5

−∂f2

∂u6

−∂f3

∂u7

−∂f4

∂u8

)

du1

+

(

∂f6

∂u1

−∂f5

∂u2

−∂f8

∂u3

+∂f7

∂u4

+∂f2

∂u5

−∂f1

∂u6

−∂f4

∂u7

+∂f3

∂u8

)

du2


+

(

∂f7

∂u1

+∂f8

∂u2

−∂f5

∂u3

−∂f6

∂u4

+∂f3

∂u5

+∂f4

∂u6

−∂f1

∂u7

−∂f2

∂u8

)

du3

+

(

∂f8

∂u1

−∂f7

∂u2

+∂f6

∂u3

−∂f5

∂u4

+∂f4

∂u5

−∂f3

∂u6

+∂f2

∂u7

−∂f1

∂u8

)

du4

+

(

∂f1

∂u1

+∂f2

∂u2

+∂f3

∂u3

+∂f4

∂u4

+∂f5

∂u5

+∂f6

∂u6

+∂f7

∂u7

+∂f8

∂u8

)

du5

−

(

∂f2

∂u1

−∂f1

∂u2

+∂f4

∂u3

−∂f3

∂u4

−∂f6

∂u5

+∂f5

∂u6

−∂f8

∂u7

+∂f7

∂u8

)

du6

−

(

∂f3

∂u1

−∂f4

∂u2

−∂f1

∂u3

+∂f2

∂u4

−∂f7

∂u5

+∂f8

∂u6

+∂f5

∂u7

−∂f6

∂u8

)

du7

−

(

∂f4

∂u1

+∂f3

∂u2

−∂f2

∂u3

−∂f1

∂u4

−∂f8

∂u5

−∂f7

∂u6

+∂f6

∂u7

+∂f5

∂u8

)

du8

]

+1

8li

[(

∂f6

∂u1

−∂f5

∂u2

−∂f8

∂u3

+∂f7

∂u4

+∂f2

∂u5

−∂f1

∂u6

−∂f4

∂u7

+∂f3

∂u8

)

du1

+

(

∂f5

∂u1

+∂f6

∂u2

+∂f7

∂u3

+∂f8

∂u4

−∂f1

∂u5

−∂f2

∂u6

−∂f3

∂u7

−∂f4

∂u8

)

du2

−

(

∂f8

∂u1

−∂f7

∂u2

+∂f6

∂u3

−∂f5

∂u4

+∂f4

∂u5

−∂f3

∂u6

+∂f2

∂u7

−∂f1

∂u8

)

du3

+

(

∂f7

∂u1

+∂f8

∂u2

−∂f5

∂u3

−∂f6

∂u4

+∂f3

∂u5

+∂f4

∂u6

−∂f1

∂u7

−∂f2

∂u8

)

du4

+

(

∂f2

∂u1

−∂f1

∂u2

+∂f4

∂u3

−∂f3

∂u4

−∂f6

∂u5

+∂f5

∂u6

−∂f8

∂u7

+∂f7

∂u8

)

du5


+

(

∂f1

∂u1

+∂f2

∂u2

+∂f3

∂u3

+∂f4

∂u4

+∂f5

∂u5

+∂f6

∂u6

+∂f7

∂u7

+∂f8

∂u8

)

du6

−

(

∂f4

∂u1

+∂f3

∂u2

−∂f2

∂u3

−∂f1

∂u4

−∂f8

∂u5

−∂f7

∂u6

+∂f6

∂u7

+∂f5

∂u8

)

du7

+

(

∂f3

∂u1

−∂f4

∂u2

−∂f1

∂u3

+∂f2

∂u4

−∂f7

∂u5

+∂f8

∂u6

+∂f5

∂u7

−∂f6

∂u8

)

du8

]

+1

8lj

[(

∂f7

∂u1

+∂f8

∂u2

−∂f5

∂u3

−∂f6

∂u4

+∂f3

∂u5

+∂f4

∂u6

−∂f1

∂u7

−∂f2

∂u8

)

du1

+

(

∂f8

∂u1

−∂f7

∂u2

+∂f6

∂u3

−∂f5

∂u4

+∂f4

∂u5

−∂f3

∂u6

+∂f2

∂u7

−∂f1

∂u8

)

du2

−

(

∂f5

∂u1

+∂f6

∂u2

+∂f7

∂u3

+∂f8

∂u4

−∂f1

∂u5

−∂f2

∂u6

−∂f3

∂u7

−∂f4

∂u8

)

du3

−

(

∂f6

∂u1

−∂f5

∂u2

−∂f8

∂u3

+∂f7

∂u4

+∂f2

∂u5

−∂f1

∂u6

−∂f4

∂u7

+∂f3

∂u8

)

du4

+

(

∂f3

∂u1

−∂f4

∂u2

−∂f1

∂u3

+∂f2

∂u4

−∂f7

∂u5

+∂f8

∂u6

+∂f5

∂u7

−∂f6

∂u8

)

du5

+

(

∂f4

∂u1

+∂f3

∂u2

−∂f2

∂u3

−∂f1

∂u4

−∂f8

∂u5

−∂f7

∂u6

+∂f6

∂u7

+∂f5

∂u8

)

du6

+

(

∂f1

∂u1

+∂f2

∂u2

+∂f3

∂u3

+∂f4

∂u4

+∂f5

∂u5

+∂f6

∂u6

+∂f7

∂u7

+∂f8

∂u8

)

du7

−

(

∂f2

∂u1

−∂f1

∂u2

+∂f4

∂u3

−∂f3

∂u4

−∂f6

∂u5

+∂f5

∂u6

−∂f8

∂u7

+∂f7

∂u8

)

du8

]


+1

8lk

[(

∂f8

∂u1

−∂f7

∂u2

+∂f6

∂u3

−∂f5

∂u4

+∂f4

∂u5

−∂f3

∂u6

+∂f2

∂u7

−∂f1

∂u8

)

du1

−

(

∂f7

∂u1

+∂f8

∂u2

−∂f5

∂u3

−∂f6

∂u4

+∂f3

∂u5

+∂f4

∂u6

−∂f1

∂u7

−∂f2

∂u8

)

du2

+

(

∂f6

∂u1

−∂f5

∂u2

−∂f8

∂u3

+∂f7

∂u4

+∂f2

∂u5

−∂f1

∂u6

−∂f4

∂u7

+∂f3

∂u8

)

du3

−

(

∂f5

∂u1

+∂f6

∂u2

+∂f7

∂u3

+∂f8

∂u4

−∂f1

∂u5

−∂f2

∂u6

−∂f3

∂u7

−∂f4

∂u8

)

du4

+

(

∂f4

∂u1

+∂f3

∂u2

−∂f2

∂u3

−∂f1

∂u4

−∂f8

∂u5

−∂f7

∂u6

+∂f6

∂u7

+∂f5

∂u8

)

du5

−

(

∂f3

∂u1

−∂f4

∂u2

−∂f1

∂u3

+∂f2

∂u4

−∂f7

∂u5

+∂f8

∂u6

+∂f5

∂u7

−∂f6

∂u8

)

du6

+

(

∂f2

∂u1

−∂f1

∂u2

+∂f4

∂u3

−∂f3

∂u4

−∂f6

∂u5

+∂f5

∂u6

−∂f8

∂u7

+∂f7

∂u8

)

du7

+

(

∂f1

∂u1

+∂f2

∂u2

+∂f3

∂u3

+∂f4

∂u4

+∂f5

∂u5

+∂f6

∂u6

+∂f7

∂u7

+∂f8

∂u8

)

du8

]

. (16)

By adequate substitution of the relations (15) in (16), one has that

h(z)dz

=1

8

{[(

∂f1

∂u1

du1+∂f1

∂u1

du1+∂f1

∂u1

du1+∂f1

∂u1

du1+∂f1

∂u1

du1+∂f1

∂u1

du1+∂f1

∂u1

du1+∂f1

∂u1

du1

)

−

(

−∂f1

∂u2

du2−∂f1

∂u2

du2−∂f1

∂u2

du2−∂f1

∂u2

du2−∂f1

∂u2

du2−∂f1

∂u2

du2−∂f1

∂u2

du2−∂f1

∂u2

du2

)

−

(

−∂f1

∂u3

du3−∂f1

∂u3

du3−∂f1

∂u3

du3−∂f1

∂u3

du3−∂f1

∂u3

du3−∂f1

∂u3

du3−∂f1

∂u3

du3−∂f1

∂u3

du3

)


−

(

−∂f1

∂u4

du4−∂f1

∂u4

du4−∂f1

∂u4

du4−∂f1

∂u4

du4−∂f1

∂u4

du4−∂f1

∂u4

du4−∂f1

∂u4

du4−∂f1

∂u4

du4

)

−

(

−∂f1

∂u5

du5−∂f1

∂u5

du5−∂f1

∂u5

du5−∂f1

∂u5

du5−∂f1

∂u5

du5−∂f1

∂u5

du5−∂f1

∂u5

du5−∂f1

∂u5

du5

)

−

(

−∂f1

∂u6

du6−∂f1

∂u6

du6−∂f1

∂u6

du6−∂f1

∂u6

du6−∂f1

∂u6

du6−∂f1

∂u6

du6−∂f1

∂u6

du6−∂f1

∂u6

du6

)

−

(

−∂f1

∂u7

du7−∂f1

∂u7

du7−∂f1

∂u7

du7−∂f1

∂u7

du7−∂f1

∂u7

du7−∂f1

∂u6

du6−∂f1

∂u7

du7−∂f1

∂u7

du7

)

−

(

−∂f1

∂u8

du8−∂f1

∂u8

du8−∂f1

∂u8

du8−∂f1

∂u8

du8−∂f1

∂u8

du8−∂f1

∂u8

du8−∂f1

∂u8

du8−∂f1

∂u8

du8

)]

+ i

[(

∂f2

∂u2

du2+∂f2

∂u2

du2+∂f2

∂u2

du2+∂f2

∂u2

du2+∂f2

∂u2

du2+∂f2

∂u2

du2+∂f2

∂u2

du2+∂f2

∂u2

du2

)

+

(

∂f2

∂u1

du1+∂f2

∂u1

du1+∂f2

∂u1

du1+∂f2

∂u1

du1+∂f2

∂u1

du1+∂f2

∂u1

du1+∂f2

∂u1

du1+∂f2

∂u1

du1

)

+

(

∂f2

∂u4

du4+∂f2

∂u4

du4+∂f2

∂u4

du4+∂f2

∂u4

du4+∂f2

∂u4

du4+∂f2

∂u4

du4+∂f2

∂u4

du4+∂f2

∂u4

du4

)

−

(

−∂f2

∂u3

du3−∂f2

∂u3

du3−∂f2

∂u3

du3−∂f2

∂u3

du3−∂f2

∂u3

du3−∂f2

∂u3

du3−∂f2

∂u3

du3−∂f2

∂u3

du3

)

−

(

−∂f2

∂u6

du6−∂f2

∂u6

du6−∂f2

∂u6

du6−∂f2

∂u6

du6−∂f2

∂u6

du6−∂f2

∂u6

du6−∂f2

∂u6

du6−∂f2

∂u6

du6

)

+

(

∂f2

∂u5

du5+∂f2

∂u5

du5+∂f2

∂u5

du5+∂f2

∂u5

du5+∂f2

∂u5

du5+∂f2

∂u5

du5+∂f2

∂u5

du5+∂f2

∂u5

du5

)

−

(

−∂f2

∂u8

du8−∂f2

∂u8

du8−∂f2

∂u8

du8−∂f2

∂u8

du8−∂f2

∂u8

du8−∂f2

∂u8

du8−∂f2

∂u8

du8−∂f2

∂u8

du8

)

+

(

∂f2

∂u7

du7+∂f2

∂u7

du7+∂f2

∂u7

du7+∂f2

∂u7

du7+∂f2

∂u7

du7+∂f2

∂u7

du7+∂f2

∂u7

du7+∂f2

∂u7

du7

)]

+ j

[(

∂f3

∂u3

du3+∂f3

∂u3

du3+∂f3

∂u3

du3+∂f3

∂u3

du3+∂f3

∂u3

du3+∂f2

∂u2

du2+∂f3

∂u3

du3+∂f3

∂u3

du3

)

−

(

−∂f3

∂u4

du4−∂f3

∂u4

du4−∂f3

∂u4

du4−∂f3

∂u4

du4−∂f3

∂u4

du4−∂f3

∂u4

du4−∂f3

∂u4

du4−∂f3

∂u4

du4

)

+

(

∂f3

∂u1

du1+∂f3

∂u1

du1+∂f3

∂u1

du1+∂f3

∂u1

du1+∂f3

∂u1

du1+∂f3

∂u1

du1+∂f3

∂u1

du1+∂f3

∂u1

du1

)


+

(

∂f3

∂u2

du2+∂f3

∂u2

du2+∂f3

∂u2

du2+∂f3

∂u2

du2 +∂f3

∂u2

du2+∂f3

∂u2

du2 +∂f3

∂u2

du2+∂f3

∂u2

du2

)

−

(

−∂f3

∂u7

du7−∂f3

∂u7

du7−∂f3

∂u7

du7−∂f3

∂u7

du7−∂f3

∂u7

du7−∂f3

∂u7

du7−∂f3

∂u7

du7−∂f3

∂u7

du7

)

+

(

∂f3

∂u8

du8+∂f3

∂u8

du8+∂f3

∂u8

du8+∂f3

∂u8

du8+∂f3

∂u8

du8+∂f3

∂u8

du8+∂f3

∂u8

du8+∂f3

∂u8

du8

)

+

(

∂f3

∂u5

du5+∂f3

∂u5

du5+∂f3

∂u5

du5+∂f3

∂u5

du5+∂f3

∂u5

du5+∂f3

∂u5

du5+∂f3

∂u5

du5+∂f3

∂u5

du5

)

−

(

−∂f3

∂u5

du5−∂f3

∂u5

du5−∂f3

∂u5

du5−∂f3

∂u5

du5−∂f3

∂u5

du5−∂f3

∂u5

du5−∂f3

∂u5

du5−∂f3

∂u5

du5

)]

+ k

[(

∂f4

∂u4

du4+∂f4

∂u4

du4+∂f4

∂u4

du4+∂f4

∂u4

du4+∂f4

∂u4

du4+∂f4

∂u4

du4+∂f4

∂u4

du4+∂f4

∂u4

du4

)

+

(

∂f4

∂u3

du3+∂f4

∂u3

du3+∂f4

∂u3

du3+∂f4

∂u3

du3+∂f4

∂u3

du3+∂f4

∂u3

du3+∂f4

∂u3

du3+∂f4

∂u3

du3

)

−

(

−∂f4

∂u2

du2−∂f4

∂u2

du2−∂f4

∂u2

du2−∂f4

∂u2

du2−∂f4

∂u2

du2−∂f4

∂u2

du2−∂f4

∂u2

du2−∂f4

∂u2

du2

)

+

(

∂f4

∂u1

du1+∂f4

∂u1

du1+∂f4

∂u1

du1+∂f4

∂u1

du1+∂f4

∂u1

du1+∂f4

∂u1

du1+∂f4

∂u1

du1+∂f4

∂u1

du1

)

−

(

−∂f4

∂u8

du8 −∂f4

∂u8

du8−∂f4

∂u8

du8−∂f4

∂u8

du8−∂f4

∂u8

du8−∂f4

∂u8

du8−∂f4

∂u8

du8−∂f4

∂u8

du8

)

−

(

−∂f4

∂u7

du7−∂f4

∂u7

du7−∂f4

∂u7

du7−∂f4

∂u7

du7−∂f4

∂u7

du7−∂f4

∂u7

du7−∂f4

∂u7

du7−∂f4

∂u7

du7

)

+

(

∂f4

∂u6

du6+∂f4

∂u6

du6+∂f4

∂u6

du6+∂f4

∂u6

du6+∂f4

∂u6

du6+∂f4

∂u6

du6+∂f4

∂u6

du6+∂f4

∂u6

du6

)

+

(

∂f4

∂u5

du5+∂f4

∂u5

du5+∂f4

∂u5

du5+∂f4

∂u5

du5+∂f4

∂u5

du5+∂f4

∂u5

du5+∂f4

∂u5

du5+∂f4

∂u5

du5

)]

+ l

[(

∂f5

∂u5

du5+∂f5

∂u5

du5+∂f5

∂u5

du5+∂f5

∂u5

du5+∂f5

∂u5

du5+∂f5

∂u5

du5+∂f5

∂u5

du5+∂f5

∂u5

du5

)

+

(

∂f5

∂u6

du6+∂f5

∂u6

du6+∂f5

∂u6

du6+∂f5

∂u6

du6+∂f5

∂u6

du6+∂f5

∂u6

du6+∂f5

∂u6

du6+∂f5

∂u6

du6

)

+

(

∂f5

∂u7

du7+∂f5

∂u7

du7+∂f5

∂u7

du7+∂f5

∂u7

du7+∂f5

∂u7

du7+∂f5

∂u7

du7+∂f5

∂u7

du7+∂f5

∂u7

du7

)


+

(

∂f5

∂u8

du8+∂f5

∂u8

du8+∂f5

∂u8

du8+∂f5

∂u8

du8+∂f5

∂u8

du8+∂f5

∂u8

du8+∂f5

∂u8

du8+∂f5

∂u8

du8

)

+

(

∂f5

∂u1

du1+∂f5

∂u1

du1+∂f5

∂u1

du1+∂f5

∂u1

du1+∂f5

∂u1

du1+∂f5

∂u1

du1+∂f5

∂u1

du1+∂f5

∂u1

du1

)

−

(

−∂f5

∂u2

du2−∂f5

∂u2

du2−∂f5

∂u2

du2−∂f5

∂u2

du2−∂f5

∂u2

du2−∂f5

∂u2

du2−∂f5

∂u2

du2−∂f5

∂u2

du2

)

−

(

−∂f5

∂u3

du3−∂f5

∂u3

du3−∂f5

∂u3

du3−∂f5

∂u3

du3−∂f5

∂u3

du3−∂f5

∂u3

du3−∂f5

∂u3

du3−∂f5

∂u3

du3

)

−

(

−∂f5

∂u7

du7−∂f5

∂u7

du7−∂f5

∂u7

du7−∂f5

∂u7

du7−∂f5

∂u7

du7−∂f5

∂u7

du7−∂f5

∂u7

du7−∂f5

∂u7

du7

)]

+ li

[(

∂f6

∂u6

du6+∂f6

∂u6

du6+∂f6

∂u6

du6+∂f6

∂u6

du6+∂f6

∂u6

du6+∂f6

∂u6

du6+∂f6

∂u6

du6+∂f6

∂u6

du6

)

−

(

−∂f6

∂u5

du5−∂f6

∂u5

du5−∂f6

∂u5

du5−∂f6

∂u5

du5−∂f6

∂u5

du5−∂f6

∂u5

du5−∂f6

∂u5

du5−∂f6

∂u5

du5

)

−

(

−∂f6

∂u8

du8−∂f6

∂u8

du8−∂f6

∂u8

du8−∂f6

∂u8

du8−∂f6

∂u8

du8−∂f6

∂u8

du8−∂f6

∂u8

du8−∂f6

∂u8

du8

)

+

(

∂f6

∂u7

du7+∂f6

∂u7

du7+∂f6

∂u7

du7+∂f6

∂u7

du7+∂f6

∂u7

du7+∂f6

∂u7

du7+∂f6

∂u7

du7+∂f6

∂u7

du7

)

+

(

∂f6

∂u2

du2+∂f6

∂u2

du2+∂f6

∂u2

du2+∂f6

∂u2

du2+∂f6

∂u2

du2+∂f6

∂u2

du2+∂f6

∂u2

du2+∂f6

∂u2

du2

)

+

(

∂f6

∂u1

du1+∂f6

∂u1

du1+∂f6

∂u1

du1+∂f6

∂u1

du1+∂f6

∂u1

du1+∂f6

∂u1

du1+∂f6

∂u1

du1+∂f6

∂u1

du1

)

−

(

−∂f6

∂u4

du4−∂f6

∂u4

du4−∂f6

∂u4

du4−∂f6

∂u4

du4−∂f6

∂u4

du4−∂f6

∂u4

du4−∂f6

∂u4

du4−∂f6

∂u4

du4

)

+

(

∂f6

∂u3

du3+∂f6

∂u3

du3+∂f6

∂u3

du3+∂f6

∂u3

du3+∂f6

∂u3

du3+∂f6

∂u3

du3+∂f6

∂u3

du3+∂f6

∂u3

du3

)]

+ lj

[(

∂f7

∂u7

du7+∂f7

∂u7

du7+∂f7

∂u7

du7+∂f7

∂u7

du7+∂f7

∂u7

du7+∂f7

∂u7

du7+∂f7

∂u7

du7+∂f7

∂u7

du7

)

+

(

∂f7

∂u8

du8 +∂f7

∂u8

du8+∂f7

∂u8

du8 +∂f7

∂u8

du8+∂f7

∂u8

du8+∂f7

∂u8

du8+∂f7

∂u8

du8+∂f7

∂u8

du8

)

−

(

−∂f7

∂u5

du5−∂f7

∂u5

du5−∂f7

∂u5

du5−∂f7

∂u5

du5−∂f7

∂u5

du5−∂f7

∂u5

du5−∂f7

∂u5

du5−∂f7

∂u5

du5

)


−

(

−∂f7

∂u6

du6−∂f7

∂u6

du6−∂f7

∂u6

du6−∂f7

∂u6

du6−∂f7

∂u6

du6−∂f7

∂u6

du6−∂f7

∂u6

du6−∂f7

∂u6

du6

)

+

(

∂f7

∂u3

du3+∂f7

∂u3

du3+∂f7

∂u3

du3+∂f7

∂u3

du3+∂f7

∂u3

du3+∂f7

∂u3

du3+∂f7

∂u3

du3+∂f7

∂u3

du3

)

+

(

∂f7

∂u4

du4+∂f7

∂u4

du4+∂f7

∂u4

du4+∂f7

∂u4

du4+∂f7

∂u4

du4+∂f7

∂u4

du4+∂f7

∂u4

du4+∂f7

∂u4

du4

)

+

(

∂f7

∂u1

du1+∂f7

∂u1

du1+∂f7

∂u1

du1+∂f7

∂u1

du1+∂f7

∂u1

du1+∂f7

∂u1

du1+∂f7

∂u1

du1+∂f7

∂u1

du1

)

−

(

−∂f7

∂u2

du2−∂f7

∂u2

du2−∂f7

∂u2

du2−∂f7

∂u2

du2−∂f7

∂u2

du2−∂f7

∂u2

du2−∂f7

∂u2

du2−∂f7

∂u2

du2

)]

+ lk

[(

∂f8

∂u8

du8+∂f8

∂u8

du8+∂f8

∂u8

du8+∂f8

∂u8

du8+∂f8

∂u8

du8+∂f8

∂u8

du8+∂f8

∂u8

du8+∂f8

∂u8

du8

)

−

(

−∂f8

∂u7

du7−∂f8

∂u7

du7−∂f8

∂u7

du7−∂f8

∂u7

du7−∂f8

∂u7

du7−∂f8

∂u7

du7−∂f8

∂u7

du7−∂f8

∂u7

du7

)

+

(

∂f8

∂u6

du6+∂f8

∂u6

du6+∂f8

∂u6

du6+∂f8

∂u6

du6+∂f8

∂u6

du6+∂f8

∂u6

du6+∂f8

∂u6

du6+∂f8

∂u6

du6

)

−

(

−∂f8

∂u5

du5−∂f8

∂u5

du5−∂f8

∂u5

du5−∂f8

∂u5

du5−∂f8

∂u5

du5−∂f8

∂u5

du5−∂f8

∂u5

du5−∂f8

∂u5

du5

)

+

(

∂f8

∂u4

du4+∂f8

∂u4

du4+∂f8

∂u4

du4+∂f8

∂u4

du4+∂f8

∂u4

du4+∂f8

∂u4

du4+∂f8

∂u4

du4+∂f8

∂u4

du4

)

−

(

−∂f8

∂u3

du3−∂f8

∂u3

du3−∂f8

∂u3

du3−∂f8

∂u3

du3−∂f8

∂u3

du3−∂f8

∂u3

du3−∂f8

∂u3

du3−∂f8

∂u3

du3

)

+

(

∂f8

∂u2

du2+∂f8

∂u2

du2+∂f6

∂u2

du2+∂f8

∂u2

du2+∂f8

∂u2

du2+∂f8

∂u2

du2+∂f8

∂u2

du2+∂f8

∂u2

du2

)

+

(

∂f8

∂u1

du1+∂f8

∂u1

du1+∂f8

∂u1

du1+∂f8

∂u1

du1+∂f8

∂u1

du1+∂f8

∂u1

du1+∂f8

∂u1

du1+∂f8

∂u1

du1

)]}

,

and to according the chain rule these terms are the total differentials of thecoordinate functions, therefore

∫

h(z)dz =

∫

(df1 + df2i + df3j + df4k + df5l + df6li + df7lj + df8lk)

= f1 + f2i + f3j + f4k + f5l + f6li + f7lj + f8lk = f(z),

what finally proofs Lemma 1.


Since the multiplication is non-comutative, there is a difference between∫

f(z)dz and∫

dzf(z), what leads to the following.

Theorem 2. For any pair of points a and b and any path joining themin a simply connected subdomain of the eight-dimensional space, the integral∫ b

adzf is independent from the given path if and only if there is a functions

G = G1 + G2i + G3j + G4k + G5l + G6li + G7lj + G8lk such that∫ b

adzf =

G(b) − G(a), and satisfying the following relations:

∂G1

∂u1

=∂G2

∂u2

=∂G3

∂u3

=∂G4

∂u4

=∂G5

∂u5

=∂G6

∂u6

=∂G7

∂u7

=∂G8

∂u8

,

∂G2

∂u1

= −∂G1

∂u2

= −∂G4

∂u3

=∂G3

∂u4

=∂G6

∂u5

= −∂G5

∂u6

=∂G8

∂u7

= −∂G7

∂u8

,

∂G3

∂u1

=∂G4

∂u2

= −∂G1

∂u3

= −∂G2

∂u4

=∂G7

∂u5

= −∂G8

∂u6

= −∂G5

∂u7

=∂G6

∂u8

,

∂G4

∂u1

= −∂G3

∂u2

=∂G2

∂u3

= −∂G1

∂u4

=∂G8

∂u5

=∂G7

∂u6

= −∂G6

∂u7

= −∂G5

∂u8

,

∂G5

∂u1

= −∂G6

∂u2

= −∂G7

∂u3

= −∂G8

∂u4

= −∂G1

∂u5

=∂G2

∂u6

=∂G3

∂u7

=∂G4

∂u8

,

∂G6

∂u1

=∂G5

∂u2

=∂G8

∂u3

= −∂G7

∂u4

= −∂G2

∂u5

= −∂G1

∂u6

=∂G4

∂u7

= −∂G3

∂u8

,

∂G7

∂u1

= −∂G8

∂u2

=∂G5

∂u3

=∂G6

∂u4

= −∂G3

∂u5

= −∂G4

∂u6

= −∂G1

∂u7

=∂G2

∂u8

,

∂G8

∂u1

=∂G7

∂u2

= −∂G6

∂u3

=∂G5

∂u4

= −∂G4

∂u5

=∂G3

∂u6

= −∂G2

∂u7

= −∂G1

∂u8

. (17)

Proof. The proof follows exactly the same steps as shown in the deductionof Theorem 1, in order to obtain:

∫ b

a

dzf =

∫

(du1 + du2i + du3j + du4k + du5l + du6li + du7lj + du8lk)

(f1 + f2i + f3j + f4k + f5l + f6li + f7lj + f8lk)

=

∫ b

a

(du1f1 − du2f2 − du3f3 − du4f4 − du5f5 − du6f6 − du7f7 − du8f8)

+

∫

b

a

(du1f2 + du2f1 + du3f4 − du4f3 − du5f6 + du6f5 − du7f8 + du8f7)i

+

∫

b

a

(du1f3 − du2f4 + du3f1 + du4f2 − du5f7 + du6f8 + du7f5 − du8f6)j

+

∫ b

a

(du1f4 + du2f3 − du3f2 + du4f1 − du5f8 − du6f7 + du7f6 + du8f5)k


+

∫ b

a

(du1f5 + du2f6 + du3f7 + du4f8 + du5f1 − du6f2 − du7f3 − du8f4)l

+

∫

b

a

(du1f6 − du2f5 − du3f8 + du4f7 + du5f2 + du6f1 − du7f4 + du8f3)li

+

∫ b

a

(du1f7 + du2f8 − du3f5 − du4f6 + du5f3 + du6f4 + du7f1 − du8f2)lj

+

∫ b

a

(du1f8 − du2f7 + du3f6 − du4f5 + du5f4 − du6f3 + du7f2 + du8f1)lk,

and is independent from the path, since

∫ b

a

dzf =

∫ b

a

dG=

∫ b

a

d(G1 + G2i + G3j + G4k + G5l + G6li + G7lj + G8lk)

= G(b) − G(a). (18)

Lemma 2. Given a function f(z) over the octonions’ quasegroup O, withproperly differentiable coordinate functions satisfying the relations in Table 1,and a function h(z) defined in terms of f(z) by

h(z) =1

8

[(

∂f1

∂u1

+∂f2

∂u2

+∂f3

∂u3

+∂f4

∂u4

+∂f5

∂u5

+∂f6

∂u6

+∂f7

∂u7

+∂f8

∂u8

)

+

(

∂f2

∂u1

−∂f1

∂u2

+∂f4

∂u3

−∂f3

∂u4

−∂f6

∂u5

+∂f5

∂u6

−∂f8

∂u7

+∂f7

∂u8

)

i

+

(

∂f3

∂u1

−∂f4

∂u2

−∂f1

∂u3

+∂f2

∂u4

−∂f7

∂u5

+∂f8

∂u6

+∂f5

∂u7

−∂f6

∂u8

)

j

+

(

∂f4

∂u1

+∂f3

∂u2

−∂f2

∂u3

−∂f1

∂u4

−∂f8

∂u5

−∂f7

∂u6

+∂f6

∂u7

+∂f5

∂u8

)

k

+

(

∂f5

∂u1

+∂f6

∂u2

+∂f7

∂u3

+∂f8

∂u4

−∂f1

∂u5

−∂f2

∂u6

−∂f3

∂u7

−∂f4

∂u8

)

l

+

(

∂f6

∂u1

−∂f5

∂u2

−∂f8

∂u3

+∂f7

∂u4

+∂f2

∂u5

−∂f1

∂u6

−∂f4

∂u7

+∂f3

∂u8

)

li


+

(

∂f7

∂u1

+∂f8

∂u2

−∂f5

∂u3

−∂f6

∂u4

+∂f3

∂u5

+∂f4

∂u6

−∂f1

∂u7

−∂f2

∂u8

)

lj

+

(

∂f8

∂u1

−∂f7

∂u2

+∂f6

∂u3

−∂f5

∂u4

+∂f4

∂u5

−∂f3

∂u6

+∂f2

∂u7

−∂f1

∂u8

)

lk

]

, (19)

then∫

dzh(z) = f(z), and hence h(z) may formally be treated as the “right-

octonion” derivative of f(z) and denoted by h(z) = dfl(z)dz

.

Proof. Repeat all steps used to demonstrate Lemma 1.

It is clear that all derivatives in the quasegroup of the octonions have thesame structure and share in common the Cauchy-Riemann relations of thecomplex variable theory,

∂f2

∂u1

= −∂f1

∂u2

,∂f1

∂u1

=∂f2

∂u2

(20)

4. Concluding Remarks

An extension of the theory of variable functions started by Hamilton has giventhe possibility for the construction of a class of systems with similar struc-ture of Cauchy-Riemann equations in a space of four dimensions (Machadoand Borges [13]). In this paper an eight dimensional analog of the Cauchy-Riemann system is studied. In introducing generalized functions as g(z), h(z),etc, defined in terms of a function f(z) of octonionic variables with coordinatefunctions obeying appropriate relations, we have obtained “hypercomplex oc-tonionic derivatives” of f(z). Further consequences of this work to others areasas to functional analysis are under investigation by our research group.

References

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Physics, 37, No. 6 (1996), 2955-2968.

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[7] R. Fueter, Du Funktionentheorie der Differentialgleichungen ∆φi = 0 und∆∆ = 0 mit vier reelen Variablen, Commet. Math. Helv., 7 (1935), 307-330.

[8] R.P. Graves, Life of Sir William Rowan Hamilton, 3 Volumes, Arno Press,New York (1975).

[9] F. Gursey, C-H. Tze, On the Role of Division, Jordon, and Related Algebras

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[10] H.R. Hamilton, Four and eight square theorems, In: Appendix 3 of The

Mathematical Papers of Sir William Rowan Hamilton, 3 (Ed-s: H. Hal-berstam, R.E. Ingram), Cambrige University Press, Cambridge (1967),648-656.

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[12] N. Jacobson, Basic Algebra I, W.H. Freeman and Company, New York(1974).

[13] J.M. Machado, M.F. Borges, New remarks on the differentiability of hy-percomplex functions, Internacional Journal of Applied Math., 8, No. 1(2002).

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